Devlin's
Angle

September 2004

A game of numbers

The approach of the 2004 World Series sees the
publication of not one but two books on the use of
statistics in baseball. By statistics, I don't mean
what most fans seem to think this means, namely
collecting and tabulating game stats, but the use of
sophisticated mathematical techniques to examine
players' performance and the effectiveness of various
plays in depth, to help clubs make hiring and salary
decisions, and to decide on game strategy.

Alan Schwartz, a senior writer at Baseball
America, has written a book called The Numbers
Game, and math professor and former MAA President
Ken Ross of the University of Oregon has published
A Mathematician at the Ballpark. These books
come close on the heels of last year's bestseller
Moneyball, by Michael Lewis, which described
how the Oakland As used mathematics to turn itself
into one of the most successful teams in the league,
despite being one of the poorest. The Schwartz book is
a history of the use of statistics in baseball; it
fills in a lot of the details that Lewis skipped over
in Moneyball. Ross's book tries to explain the
math itself. All the facts in this article are taken
from one or more of these three books.

At this point, I need to admit up front that I am not
a baseball fan. I attended my first Major League game
only this year. Not that I have anything against the
game. Just that, growing up in England, baseball
looked to me like rounders played by men in pyjamas
who seemed to wear very scratchy underpants that
required constant adjustment and who had an unusual
propensity for spitting.

Still, as Alan Schwartz points out in his book, many
thousands of Americans got interested in math by
collecting baseball statistics, including some who
went on to be professors of mathematics at major
universities like Harvard. I may be one of the few
people in the world who did it the other way: I have
become interested in baseball (to a degree) through
math. And in writing about baseball, as I am now
doing, I am following in the tradition set by the
"father of baseball": Henry Chadwick.

Chadwick was a young English cricket reporter who
became interested in baseball in 1856. (The game
itself evolved from the English games of cricket and
rounders in the 1830s.) Throughout a long career as a
sports writer, Chadwick was an avid (and highly
opinionated) promoter of the collection of baseball
statistics and the computation of various measures of
player and team performance, including the
subsequently famous - though not particularly
informative it turns out - batting average.

Moneyball

In 1997, when a former player called Billy Beane
became General Manager of the Oakland Athletics, the
team was one of the worst in the game, ending the year
with 65 wins and 97 losses, 25 games behind the
Seattle Mariners, who won their division. Beane's
problem in addressing this weakness was that the As
was also one of the poorest teams. While the New York
Yankees spent $126 million on its twenty-five players
that year, and had another $100 million to dip into if
needed, Beane had just $40 million. (Exact financial
comparisons are impossible, since clubs organize and
report their finances in different ways, but you get
some idea of the financial differences from the fact
that for the period 1995-99, the Oakland As reported a
loss of $44.95 million while the Yankees declared a
profit of $64.5 million.) With more money, Beane might
have gone shopping for some All-Star players, or else
sent out his scouts to find some untapped talent in a
high school or college, or perhaps overseas.

Beane did neither. His first major hire for the
Athletics was a 26-year-old assistant manager named
Paul DePodesta, who had majored in economics at
Harvard. DePodesta had never been much of an athlete.
As a reserve infielder on the university's baseball
team, he "couldn't run, couldn't throw and had no
power," he later told reporters, and as a wide
receiver for the football team he had quickly come to
realize that "the sideline was my friend." DePodesta
could do one thing well, though, and that was enough:
Mathematics.

What happened next is the subject of Moneyball.
By surfing the Internet, downloading baseball
data, and using statistical software to analyze it,
DePodesta managed to put together a winning team for
a fraction of what his competitors spent.

The situation is reminiscent of Wall Street in the
1980s, when a group of young mathematical types began
to apply their skills to the stock market. Instead of
relying on experience, intuition, foresight, or other
traditional "people skills," the newcomers treated the
market in purely abstract terms, as an enormous equation.
And they made a killing.

Baseball is particularly suited to a by-the-numbers
approach, Beane and DePodesta realized, because it's
so dependent on individual performances. In football,
a play can only succeed if all players on a team work
together - blocking, running, catching, and throwing
in synch. But Barry Bonds doesn't need anyone's help
to hit a home run. In baseball, every pitch, every
hit, every catch or throw depends on an individual's
success or failure, so it can be given a precise
numerical value. Take those numbers and analyze them
dispassionately and you just might have the makings of
a championship team.

For example, in 2001, DePodesta tried to draft a
college player called Kevin Youkilis, an overweight
third baseman who could neither run, throw, or field.
The one thing he did have was the second highest
on-base percentage in all of baseball after Barry
Bonds. Similarly, no other Major League team had
shown any interest in Jeremy Brown, a senior catcher
at the University of Alabama. To the traditional
scouts, Brown, with a soft, chubby body, simply did
not look like someone who could play ball. To
DePodesta, he was a player who racked up an awesome
number of walks, and the math told DePodesta that
walks were supremely important to winning games. The
As made Brown a first-round draft pick in 2001.

By reducing baseball to a numbers game in this way,
the Oakland As confounded all the old-time baseball
pundits, becoming champions of the Western division of
the American League in 2000, 2002, 2003, and setting
an American League record by winning 20 consecutive
games in 2002.

Patterns galore

As Schwartz makes clear in his book, baseball has
always been, in one way or another, a game of numbers,
and keeping statistics was viewed as important from
the very start. The first primitive box score was
printed in the New York Morning News in 1845.
Soon after that, newspapers regularly printed tables
of statistics after each game.

[Since this column is read by people with little
background in mathematics, at this juncture maybe I
should note that statisticians talk about two kinds of
statistics: little-s statistics and capital-S
statistics. Little-s statistics is (or are) numbers:
counting, tabulating, calculating averages, and so on.
Big-S statistics is the use of (often advanced)
mathematics to process all those numbers in order to
make informed decisions: such as, whether to hire
player A or player B and for how much, whether batting
order is really important (no), whether there is
really such a thing as a clutch hitter (no), or
whether Joe DiMaggio's 56-game hitting streak was just
a matter of luck (no, although many other hallowed
records probably were), etc.]

Baseball is a natural game both to collect little-s
statistics in and to apply big-S Statistics to. One
reason is obvious: there are lots of things to count.
Another reason may be a bit less obvious: for all the
skill and artistry of the great players, there is an
enormous random element to the game. From those two
observations, it doesn't take much mathematical
knowledge to realize that it is likely to be quite
hard to separate mirages from reality.

There have been over 11 million batter-pitcher
confrontations in the more than 150,000 games that
have been played since the major leagues began. With
so much raw data and so many things you can do with
that data, coupled with a big random element, you are
going to get lots of patterns. Figuring out whether
they tell you anything useful is likely to be very
difficult. As several experts have observed, many of
the most hallowed streaks and other records that made
players famous are quite likely simply the result of
pure luck. Like winning the lottery, sooner or later
one player or another would have done it.

For instance, if the outcome of every play in major
league history had been decided purely on luck, with
no skill involved, someone would have chalked up a
.424 batting average (as Rogers Hornsby did), someone
would have scored three home runs in a World Series
game (as Babe Ruth and Reggie Jackson did), and a
whole ton of players would have earned a reputation
for being clutch hitters (as many did). The only
record that would not have arisen through pure luck is
DiMaggio's 56-game hitting streak. The best that would
have happened by chance is a 46-game streak, or
thereabouts.

This does not mean that there isn't a lot of skill
involved in baseball. Nor does it mean that some
players aren't better than others. It does suggest
that there is a lot more that happens due to pure luck
than most fans (or record holding players) would like
to admit.

But what statistical theory taketh away with one hand,
it can give back, at least in part, with the other. A
good example of what can be done with advanced
statistics is a 1997 study made by Harvard
statistician Carl Morris of the legendary Ty Cobb's
batting record. Cobb hit above .400 for three seasons,
.420 in 1911, .409 in 1912, and .401 in 1922. The
question then is: was Cobb a true .400 hitter? He
might have been a .385 hitter who got lucky those
three seasons. On the other hand, he was just below
.400 for two seasons, .390 in 1913 and .389 in 1921,
so maybe he was a .400 hitter who just got unlucky
those years. Morris analyzed Cobb's entire record and
concluded that there is an 88% chance that Cobb was a
true .400 hitter for some season (though not
necessarily one of the three seasons when he actually
hit that level).

How do you measure how good a batter is?

Ross, in his book, explains the most common baseball
statistics for evaluating batters. By far the most
common, although the least informative of all, is
batting average, which, according to Schwartz, a man
called H. A. Dobson of Washington suggested in a
letter to Chadwick, who thereafter promoted it in his
writings. Batting average, AVG, for a given period, is
given by dividing the number of hits, H, by the number
of official at-bats, AB:

AVG = H/AB

Batting averages are now generally regarded as a poor
guide to performance - not least because they do not
distinguish between a single, double, triple, or home
run, or how many players on bases advance by virtue of
a hit. They are also mathematically problematic, as
shown by a curious phenomenon that can crop up known
as Simpson's paradox, which Ross describes in his
book.

Consider the records for Major League players Derek
Jeter and David Justice in 1995 and 1996.

In 1995, Jeter had 12 hits from 48 at bats for an
average of .250, while Justice had 104 hits from 411
at bats for an average of .253. So in 1995, Justice
looks better than Jeter.

In 1996, Jeter had 183 hits from 582 at bats for an
average of .314, while Justice had 45 hits from 140 at
bats for an average of .321. Again, Justice looks
better than Jeter.

So who was the better hitter over the two year period
combined? You might think it is Justice. After all, in
each year, Justice had the higher average. But do the
math.

For the two year period 1995-96 combined, Jeter had 12
+ 183 = 195 hits from 48 + 582 = 630 at bats for an
average of .310, while Justice had 104 + 45 = 149 hits
from 411 + 140 = 551 at bats for an average of .270.
So over the two year period, Jeter did much better
than Justice. Curious, no? Just who was the better
hitter? Batting average won't tell you.

As I mentioned a moment ago, the most obvious
defficiency of batting average is that it ignores
extra-base hits, runs batted in, and bases
on balls. A better statistic, that came to prominence
in the 1980s, is slugging percentage, SLG. This takes
into account the total number of bases, TB, given by:

TB = 1B + 2 x 2B + 3 x 3B + 4 x HR

where 1B is the number of singles, 2B the number of
doubles, 3B the number of triples, and HR the number
of home runs. The slugging percentage is given by the
formula:

SLG = TB/AB

Although better than batting average, slugging
percentage (which, as defined, is not a percentage,
although the answer can easily be given as one), is
problematic in that it gives too much weight to
extra-base hits.

These days, arguably the most popular measure of
batter effectiveness - because it has been shown to be
very accurate - is the on-base percentage, OBP. This
gives the proportion of actual plate appearances where
the player gets on base (or scores a home run). It is
given by

OBP = [H + BB + HBP]/PA

where BB is the number of bases on balls, HBP is the
number of times the batter is hit by a pitched ball,
and PA is the number of plate appearances, given by

PA = AB + BB + HBP + SF

where SF is the number of times the batter hits a
sacrifice fly.

OBP was introduced in Sports Illustrated in
1956, which reported that Duke Snider of Brooklyn had
led the National League with a 39.94 on-base
percentage. In the 1960s, baseball statistician Pete
Palmer ran correlation analyses that showed OBP was
far superior to batting average and slightly more
important than the more widely known slugging
percentage. By then, the more statistically savvy fans
had realized something that few of their fellow fans,
and apparently few managers, knew: avoiding outs,
which OBP measures, was more important than hitting
runs. One person who did realize this was a man called
Eric Walker, of whom more later.

An even better measure of performance than slugging
percentage or on-base percentage is their sum, known
rather imaginatively as on-base plus slugging:

OPS = SLG + OPB

In the late 1970s, a self-styled baseball writer
called Bill James (we'll meet him again later)
discovered a remarkable statistic for measuring a
batter's performance that he called the "Runs Created
Formula":

RC = (H + BB) x (Total bases)/[AB + BB]

This formula, which James discovered by trial and
error, turns out to be a remarkably accurate predictor
of the total number of runs a team will make in a
season. Consequently, the higher the RC value, the
more games the team will win overall. (The formula
won't tell you much about winning the World Series,
since that depends on the outcome of a small number of
specific games; rather, like all statistical
techniques, its accuracy is over a complete season.)
The value of the RC formula to the team manager is
that it shows the exact relative importance of the
contributions players with different talents can make
to a team's overall performance. It shows, for
example, that walks are a major contribution to a
team's overall success, whereas batting averages, by
not figuring in the formula, are largely irrelevant.

A brief history of baseball statistics

Following Chadwick, a major impetus to the collection
and tabulation of statistics in baseball came with
Babe Ruth's exploits in the 1920s. With Ruth, the
focus shifted clearly from team performance to
individual performance.

There were many errors in the collection and recording
of statistics, some of which were not discovered until
many decades later. For instance, Ty Cobb is credited
with a .401 batting average in 1922, but the true
figure is now known to be .399. This particular
discrepancy was discovered at the time, but ignored to
avoid annoying fans by lowering the record below the
magic .400.

In 1951, the Official Encyclopedia of Baseball
was published, listing (for the first time) every
major league player, past or present, with the batting
averages given for each hitter and the won-lost record
for each pitcher.

In the late 1950s and early 1960s, George Lindsey, an
Operations Research expert at the Canadian Department
of Defense, applied Operations Research techniques to
analyze past games and develop baseball strategies.
For instance, he found that the sacrifice bunt has
value only late in the game when just one run is
needed, and that stealing bases is rarely worth it. He
also found that when hitters faced pitchers of the
opposite handedness, batting averages go up by 32
points and that a true .300 hitter would often bat
.180 over as many as seven games due purely to
randomness. No one outside the OR and statistics
communities took any notice.

A similar fate met the efforts of several others
statisticians and OR practitioners.

The appearance in 1964 of the book Percentage
Baseball, by Earnshaw Cook, drew more widespread
attention, but still had little impact on clubs. Using
a statistic called Scoring Index, Cook showed that the
best ever hitter was Ty Cobb, beating out Babe Ruth,
Ted Williams, and Lou Gehrig. This statistic was way
ahead of its time. It's very close to the modern
on-base percentage times slugging percentage.

In 1965, David Neft, a statistician for the Lou Harris
polling organization with a BA in Statistics from
Columbia University, persuaded Information Concepts,
Inc. to commission him to create a computerized
baseball encyclopedia. Neft soon realized that the
existing records were so error ridden, he would have
to recreate all of baseball's statistical record from
the very beginnings of the game. He hired a staff of
21 researchers to work on the task for two years,
traveling all over the country looking at original
game reports in newspapers. The book was published in
1969. It had 2,338 pages and weighed six-and-a-half
pounds. It came out to both rave reviews and
controversy - the latter because it corrected many
hallowed records. Its appearance also led to the
formation in 1971 of SABR, the Society for American
Baseball Research.

The initial group of 16 professional statisticians who
gathered in Cooperstown, New York, in 1971, to form
SABR has grown today to over 7,000 members worldwide,
and produces an annual journal, The Baseball
Research Journal. From the start, the SABR
statisticians were less interested in ranking players,
than in improving overall play. They made use of the
latest statistical techniques, coupled these days with
masses of computing power.

Very few SABR members are in professional baseball.
The organization includes some sports journalists, but
for the most part the members' love of baseball is
purely an amateur one, albeit pursued with
professional zeal. What they do bring to the game is a
wealth of knowledge, ability and experience in the
application of statistical techniques. (In the early
days, before the advent of powerful desktop computers,
they also brought access to some of the nation's most
sophisticated computer systems - the systems of their
employers, which were set to work on baseball
statistics during the night, sometimes in secret, when
the company was not using them.)

In 1977, a self-styled sports writer named Bill James
published the first of what would become an annual
(and initially self-published) magazine: Baseball
Abstract, which ran until 1988. In it, in addition
to saying some remarkably sensible things about
baseball statistics, James coined the term
"sabermetrics" to refer to the application of
mathematical principles to the production and use of
statistics in baseball, as advocated and carried out
by the members of SABR.

James was particularly critical of the statistical
measures advocated by Henry Chadwick, among them
fielding errors, batting average, and RBI (runs batted
in). Those statistics were easy to understand and to
calculate, so baseball managers and coaches took to
them right away. But they were often less informative
than they appeared. For example, Chadwick measured a
fielder's performance by his number of errors, yet to
have an event recorded as an error the fielder has to
do something right by being in the right place at the
right time, and what he does is an "error" only when
the observer makes a judgment of what another fielder
might have done in the same circumstances. Chadwick
also recorded a walk as a pitcher's error but gave no
credit to the hitter who might have shown great
judgment in deciding when to swing.

By a process of trial and error, James also produced
his now famous Runs Created Formula, which we met
earlier, and which is a remarkably good predictor of a
team's overall success.

In 1981, a data company called STATS, Inc. was formed,
securing a major contract to supply statistics for
reporters covering the Oakland As, who wanted to
increase their fan base. Soon after, the Chicago White
Sox signed up to secure data for salary negotiations.
Despite this encouraging start, four years later the
company was effectively bankrupt; the baseball world
was not yet ready for a computerized statistical
service.

By the late 1980s, efforts by SABR members had
uncovered many errors in the Baseball
Encylopedia. (According to Schwartz, many of them
were introduced deliberately by the editor who took
over from Neft, Joe Reichler, who wanted to revert to
the older, but incorrect records that Neft's team had
corrected.)

In 1989, Pete Palmer and John Thorn published their
book Total Baseball, a 2,294 page volume that
not only put the records straight but also gave many
of the newer statistics that had been developed,
including James's Runs Created.

In 1990, STATS, Inc. was still in existence - just -
due to a collaboration with Bill James' amateur
network Project Scoresheet, which collected game
statistics through a nationwide network of amateurs.
That year, the company secured a major contract with
USA Today to supply the statistics for a
massively revamped daily box score. It was back in
business. The same year, Electronic Arts bought STATS,
Inc. data for its Earl Weaver Baseball game,
and ESPN used STATS to supply the statistics for its
newly launched MLB coverage.

In 1991, Associated Press abandoned their own data
collection organization and contracted to buy their
statistics from STATS, Inc. and in 1994, STATS went
live on the Internet, supplying detailed numbers of
games as they were being played.

What Moneyball left out

By focusing on Beane and DePodesta's efforts at the
Oakland As, Lewis's account in Moneyball gives
two impressions that, according to Schwartz, are
misleading.

First, the Oakland As was not the first club to make a
serious attempt to use statistics to build a team. The
Brooklyn Dodgers had tried it in the late 1940s and
early 50s. Manager Branch Rickey made the first ever
hire of a full-time statistician, Allan Roth, and the
two of them set out to apply mathematics to building a
baseball team. For example:

Based on Roth's statistical analyses, after the
1947 season, in which Dixie Walker had hit .306, he
was traded to Pittsburgh. Two years later he was out
of the majors, his form having completely gone, as
Roth's figures had indicated would happen.

In 1949, Jackie Robinson, a lowly .296 hitter, was
moved to the cleanup spot. Why? Because Roth's figures
showed that Robinson had batted .350 with men on base.
By the end of the year, Robinson was named National
League MVP, batting .342 with 124 RBIs.

In May 1952, Roy Campanella was batting .325, but
the team played Rube Walker against Cincinnati, a
player in the low .200s. Why? Roth's figures showed
that Campanella's liftetime batting average against the
Reds' pitcher was a paltry .065.

The first manager known to have used statistics in the
dugout was Earl Weaver, manager of the Orioles from
1968 to 1982. He kept up-to-date player statistics on
index cards and consulted them in making play
decisions.

Lewis's second false impression, according to
Schwartz, is that Beane introduced the mathematical
approach to the As. The honor for that, according to
Schwartz, goes to an NPR sports reporter called Eric
Walker, way back in 1981. Schwartz tells the whole
story.

In the 1970s, Walker, a former aerospace engineer, had
started doing some radio reporting for the NPR
affiliate KQED in San Francisco. A chance visit to a
San Francisco Giants game was all it took for him to
see through his engineer's eyes what few others
appeared to have noticed: the supreme importance of
walks and of a batter not getting out.

Walker started to talk about his observation, and some
ideas to capitalize on it, in his daily five-minute
morning baseball report on NPR.

He also took his ideas to the Giants, but they never
bought into them, so a few months later he took his
pitch across the Bay to the Oakland As. By then, he
had written up his ideas in a little book called
The Sinister First Baseman. The As' legal
counsel, Sandy Alderson, had heard Walker on NPR, and
had just read his book, and as a result Walker was
very well received by the club. The As hired him as a
consultant, and started to implement his ideas. Among
the decisions they made by following Walker's creed
were:

In June 1984, they drafted slugger Mark McGwire
tenth overall rather than two more speed-oriented
players, Shane Mack and Oddibe McDowell.

In 1986, they let go slugger Dave Kingman (35
homers and 94 RBIs the previous season) because he
rarely walked, having a low OBP of .258. They signed
Reggie Jackson (OBP .381) as new designated hitter.

In 1987 they traded Alfredo Griffin, who drew few
walks, for pitcher Bob Welch.

In 1988 they signed Don Baylor, a power hitter who
frequently got on base by being hit by the ball.

In 1989, they acquired Rickey Henderson, who had a
super record of both homers and walks, and Ken Phelps,
another player with a high OBP, both from the Yankees.

In 1990 they acquired another good walker, Harold
Baines.

By concentrating on OBP, the As became a highly
successful team, winning four division titles and
three American League pennants from 1988 to 1992.
Then they hired Billy Beane, and started to
indoctrinate him in their ways.

Devlin's Angle is updated at the
beginning
of each month.
Mathematician Keith Devlin (
devlin@csli.stanford.edu) is the
Executive Director of the Center for the
Study of Language and Information at
Stanford University and
The Math Guy on NPR's Weekend Edition.
Devlin's newest book, THE MATH INSTINCT:
The Amazing Mathematical Abilities of Animals
and All of Us, will be published next spring
by Thunder's Mouth Press.